Lloyds sealed its preferred bidder status having seen off rivals Santander UK and Royal Bank of Scotland, reported Sky News.
The price for the book was pegged in the tens of millions of pounds.
The exclusive talks arrangement will be in place until the end of August.
Lloyds is due to release its H1 2019 trading update on 31 July.
The news is likely to ease fears among campaigners and MPs who have urged Tesco to sell its book to an active lender.
Tesco Bank announced it was ceasing mortgage lending in May, closing its book of 23,000 customers.